Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, E

Percentage Accurate: 99.9% → 99.9%

Time: 9.2s

Alternatives: 9

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \left(1 - x\right) + y \cdot \sqrt{x} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ (- 1.0 x) (* y (sqrt x))))

C

double code(double x, double y) {
	return (1.0 - x) + (y * sqrt(x));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (1.0d0 - x) + (y * sqrt(x))
end function

Java

public static double code(double x, double y) {
	return (1.0 - x) + (y * Math.sqrt(x));
}

Python

def code(x, y):
	return (1.0 - x) + (y * math.sqrt(x))

Julia

function code(x, y)
	return Float64(Float64(1.0 - x) + Float64(y * sqrt(x)))
end

MATLAB

function tmp = code(x, y)
	tmp = (1.0 - x) + (y * sqrt(x));
end

Wolfram

code[x_, y_] := N[(N[(1.0 - x), $MachinePrecision] + N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(1 - x\right) + y \cdot \sqrt{x} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ (- 1.0 x) (* y (sqrt x))))

C

double code(double x, double y) {
	return (1.0 - x) + (y * sqrt(x));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (1.0d0 - x) + (y * sqrt(x))
end function

Java

public static double code(double x, double y) {
	return (1.0 - x) + (y * Math.sqrt(x));
}

Python

def code(x, y):
	return (1.0 - x) + (y * math.sqrt(x))

Julia

function code(x, y)
	return Float64(Float64(1.0 - x) + Float64(y * sqrt(x)))
end

MATLAB

function tmp = code(x, y)
	tmp = (1.0 - x) + (y * sqrt(x));
end

Wolfram

code[x_, y_] := N[(N[(1.0 - x), $MachinePrecision] + N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(1 - x\right) + y \cdot \sqrt{x} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ (- 1.0 x) (* y (sqrt x))))

C

double code(double x, double y) {
	return (1.0 - x) + (y * sqrt(x));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (1.0d0 - x) + (y * sqrt(x))
end function

Java

public static double code(double x, double y) {
	return (1.0 - x) + (y * Math.sqrt(x));
}

Python

def code(x, y):
	return (1.0 - x) + (y * math.sqrt(x))

Julia

function code(x, y)
	return Float64(Float64(1.0 - x) + Float64(y * sqrt(x)))
end

MATLAB

function tmp = code(x, y)
	tmp = (1.0 - x) + (y * sqrt(x));
end

Wolfram

code[x_, y_] := N[(N[(1.0 - x), $MachinePrecision] + N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left(1 - x\right) + y \cdot \sqrt{x} \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 2: 95.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + y \cdot \sqrt{x}\\ \mathbf{if}\;y \leq -6.8 \cdot 10^{+46}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{+49}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* y (sqrt x)))))
   (if (<= y -6.8e+46) t_0 (if (<= y 2.3e+49) (- 1.0 x) t_0))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (y * sqrt(x));
	double tmp;
	if (y <= -6.8e+46) {
		tmp = t_0;
	} else if (y <= 2.3e+49) {
		tmp = 1.0 - x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (y * sqrt(x))
    if (y <= (-6.8d+46)) then
        tmp = t_0
    else if (y <= 2.3d+49) then
        tmp = 1.0d0 - x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + (y * Math.sqrt(x));
	double tmp;
	if (y <= -6.8e+46) {
		tmp = t_0;
	} else if (y <= 2.3e+49) {
		tmp = 1.0 - x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (y * math.sqrt(x))
	tmp = 0
	if y <= -6.8e+46:
		tmp = t_0
	elif y <= 2.3e+49:
		tmp = 1.0 - x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(y * sqrt(x)))
	tmp = 0.0
	if (y <= -6.8e+46)
		tmp = t_0;
	elseif (y <= 2.3e+49)
		tmp = Float64(1.0 - x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + (y * sqrt(x));
	tmp = 0.0;
	if (y <= -6.8e+46)
		tmp = t_0;
	elseif (y <= 2.3e+49)
		tmp = 1.0 - x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -6.8e+46], t$95$0, If[LessEqual[y, 2.3e+49], N[(1.0 - x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -6.7999999999999996e46 or 2.30000000000000002e49 < y

   1. Initial program 99.8%

      \[\left(1 - x\right) + y \cdot \sqrt{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + \sqrt{x} \cdot y} \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\sqrt{x} \cdot y\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{y}\right)\right) \]

      3. sqrt-lowering-sqrt.f64 94.0%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), y\right)\right) \]

   5. Simplified 94.0%

      \[\leadsto \color{blue}{1 + \sqrt{x} \cdot y} \]

   if -6.7999999999999996e46 < y < 2.30000000000000002e49

   1. Initial program 100.0%

      \[\left(1 - x\right) + y \cdot \sqrt{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 - x} \]
   4. Step-by-step derivation

      1. --lowering--.f64 98.6%

         \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{x}\right) \]

   5. Simplified 98.6%

      \[\leadsto \color{blue}{1 - x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.8 \cdot 10^{+46}:\\ \;\;\;\;1 + y \cdot \sqrt{x}\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{+49}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \sqrt{x}\\ \mathbf{if}\;y \leq -2.8 \cdot 10^{+48}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 5.4 \cdot 10^{+76}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (sqrt x))))
   (if (<= y -2.8e+48) t_0 (if (<= y 5.4e+76) (- 1.0 x) t_0))))

C

double code(double x, double y) {
	double t_0 = y * sqrt(x);
	double tmp;
	if (y <= -2.8e+48) {
		tmp = t_0;
	} else if (y <= 5.4e+76) {
		tmp = 1.0 - x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y * sqrt(x)
    if (y <= (-2.8d+48)) then
        tmp = t_0
    else if (y <= 5.4d+76) then
        tmp = 1.0d0 - x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y * Math.sqrt(x);
	double tmp;
	if (y <= -2.8e+48) {
		tmp = t_0;
	} else if (y <= 5.4e+76) {
		tmp = 1.0 - x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * math.sqrt(x)
	tmp = 0
	if y <= -2.8e+48:
		tmp = t_0
	elif y <= 5.4e+76:
		tmp = 1.0 - x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * sqrt(x))
	tmp = 0.0
	if (y <= -2.8e+48)
		tmp = t_0;
	elseif (y <= 5.4e+76)
		tmp = Float64(1.0 - x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * sqrt(x);
	tmp = 0.0;
	if (y <= -2.8e+48)
		tmp = t_0;
	elseif (y <= 5.4e+76)
		tmp = 1.0 - x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.8e+48], t$95$0, If[LessEqual[y, 5.4e+76], N[(1.0 - x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.80000000000000012e48 or 5.3999999999999998e76 < y

   1. Initial program 99.7%

      \[\left(1 - x\right) + y \cdot \sqrt{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\sqrt{x} \cdot y} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{y}\right) \]

      2. sqrt-lowering-sqrt.f64 90.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), y\right) \]

   5. Simplified 90.3%

      \[\leadsto \color{blue}{\sqrt{x} \cdot y} \]

   if -2.80000000000000012e48 < y < 5.3999999999999998e76

   1. Initial program 100.0%

      \[\left(1 - x\right) + y \cdot \sqrt{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 - x} \]
   4. Step-by-step derivation

      1. --lowering--.f64 96.9%

         \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{x}\right) \]

   5. Simplified 96.9%

      \[\leadsto \color{blue}{1 - x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{+48}:\\ \;\;\;\;y \cdot \sqrt{x}\\ \mathbf{elif}\;y \leq 5.4 \cdot 10^{+76}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \sqrt{x}\\ \mathbf{if}\;x \leq 0.0042:\\ \;\;\;\;1 + t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_0 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (sqrt x)))) (if (<= x 0.0042) (+ 1.0 t_0) (- t_0 x))))

C

double code(double x, double y) {
	double t_0 = y * sqrt(x);
	double tmp;
	if (x <= 0.0042) {
		tmp = 1.0 + t_0;
	} else {
		tmp = t_0 - x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y * sqrt(x)
    if (x <= 0.0042d0) then
        tmp = 1.0d0 + t_0
    else
        tmp = t_0 - x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y * Math.sqrt(x);
	double tmp;
	if (x <= 0.0042) {
		tmp = 1.0 + t_0;
	} else {
		tmp = t_0 - x;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * math.sqrt(x)
	tmp = 0
	if x <= 0.0042:
		tmp = 1.0 + t_0
	else:
		tmp = t_0 - x
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * sqrt(x))
	tmp = 0.0
	if (x <= 0.0042)
		tmp = Float64(1.0 + t_0);
	else
		tmp = Float64(t_0 - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * sqrt(x);
	tmp = 0.0;
	if (x <= 0.0042)
		tmp = 1.0 + t_0;
	else
		tmp = t_0 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 0.0042], N[(1.0 + t$95$0), $MachinePrecision], N[(t$95$0 - x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.00419999999999999974

   1. Initial program 99.9%

      \[\left(1 - x\right) + y \cdot \sqrt{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + \sqrt{x} \cdot y} \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\sqrt{x} \cdot y\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{y}\right)\right) \]

      3. sqrt-lowering-sqrt.f64 98.9%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), y\right)\right) \]

   5. Simplified 98.9%

      \[\leadsto \color{blue}{1 + \sqrt{x} \cdot y} \]

   if 0.00419999999999999974 < x

   1. Initial program 99.9%

      \[\left(1 - x\right) + y \cdot \sqrt{x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(1, x\right), \left(\sqrt{x} \cdot \color{blue}{y}\right)\right) \]

      2. pow1/2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(1, x\right), \left({x}^{\frac{1}{2}} \cdot y\right)\right) \]

      3. sqr-pow N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(1, x\right), \left(\left({x}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{2}}{2}\right)}\right) \cdot y\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(1, x\right), \left({x}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot \color{blue}{\left({x}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot y\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{*.f64}\left(\left({x}^{\left(\frac{\frac{1}{2}}{2}\right)}\right), \color{blue}{\left({x}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot y\right)}\right)\right) \]

      6. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, \left(\frac{\frac{1}{2}}{2}\right)\right), \left(\color{blue}{{x}^{\left(\frac{\frac{1}{2}}{2}\right)}} \cdot y\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, \frac{1}{4}\right), \left({x}^{\color{blue}{\left(\frac{\frac{1}{2}}{2}\right)}} \cdot y\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, \frac{1}{4}\right), \mathsf{*.f64}\left(\left({x}^{\left(\frac{\frac{1}{2}}{2}\right)}\right), \color{blue}{y}\right)\right)\right) \]

      9. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, \frac{1}{4}\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, \left(\frac{\frac{1}{2}}{2}\right)\right), y\right)\right)\right) \]

      10. metadata-eval 99.7%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, \frac{1}{4}\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, \frac{1}{4}\right), y\right)\right)\right) \]

   4. Applied egg-rr 99.7%

      \[\leadsto \left(1 - x\right) + \color{blue}{{x}^{0.25} \cdot \left({x}^{0.25} \cdot y\right)} \]

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\sqrt{\frac{1}{x}} \cdot y - 1\right)} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\sqrt{\frac{1}{x}} \cdot y - 1\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\sqrt{\frac{1}{x}} \cdot y + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\sqrt{\frac{1}{x}} \cdot y + -1\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\sqrt{\frac{1}{x}} \cdot y\right), \color{blue}{-1}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(y \cdot \sqrt{\frac{1}{x}}\right), -1\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \left(\sqrt{\frac{1}{x}}\right)\right), -1\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{sqrt.f64}\left(\left(\frac{1}{x}\right)\right)\right), -1\right)\right) \]

      8. /-lowering-/.f64 99.3%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right)\right), -1\right)\right) \]

   7. Simplified 99.3%

      \[\leadsto \color{blue}{x \cdot \left(y \cdot \sqrt{\frac{1}{x}} + -1\right)} \]

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-1 \cdot x + \sqrt{x} \cdot y} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \sqrt{x} \cdot y + \color{blue}{-1 \cdot x} \]

      2. mul-1-neg N/A

         \[\leadsto \sqrt{x} \cdot y + \left(\mathsf{neg}\left(x\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \sqrt{x} \cdot y - \color{blue}{x} \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\sqrt{x} \cdot y\right), \color{blue}{x}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{x}\right), y\right), x\right) \]

      6. sqrt-lowering-sqrt.f64 99.5%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), y\right), x\right) \]

   10. Simplified 99.5%

       \[\leadsto \color{blue}{\sqrt{x} \cdot y - x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0042:\\ \;\;\;\;1 + y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \sqrt{x} - x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.1% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.15 \cdot 10^{+91}:\\ \;\;\;\;1 + x \cdot \left(-1 - y \cdot y\right)\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{+93}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;1 - \left(x - y \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3.15e+91)
   (+ 1.0 (* x (- -1.0 (* y y))))
   (if (<= y 1.8e+93) (- 1.0 x) (- 1.0 (- x (* y y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3.15e+91) {
		tmp = 1.0 + (x * (-1.0 - (y * y)));
	} else if (y <= 1.8e+93) {
		tmp = 1.0 - x;
	} else {
		tmp = 1.0 - (x - (y * y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-3.15d+91)) then
        tmp = 1.0d0 + (x * ((-1.0d0) - (y * y)))
    else if (y <= 1.8d+93) then
        tmp = 1.0d0 - x
    else
        tmp = 1.0d0 - (x - (y * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -3.15e+91) {
		tmp = 1.0 + (x * (-1.0 - (y * y)));
	} else if (y <= 1.8e+93) {
		tmp = 1.0 - x;
	} else {
		tmp = 1.0 - (x - (y * y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -3.15e+91:
		tmp = 1.0 + (x * (-1.0 - (y * y)))
	elif y <= 1.8e+93:
		tmp = 1.0 - x
	else:
		tmp = 1.0 - (x - (y * y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3.15e+91)
		tmp = Float64(1.0 + Float64(x * Float64(-1.0 - Float64(y * y))));
	elseif (y <= 1.8e+93)
		tmp = Float64(1.0 - x);
	else
		tmp = Float64(1.0 - Float64(x - Float64(y * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -3.15e+91)
		tmp = 1.0 + (x * (-1.0 - (y * y)));
	elseif (y <= 1.8e+93)
		tmp = 1.0 - x;
	else
		tmp = 1.0 - (x - (y * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3.15e+91], N[(1.0 + N[(x * N[(-1.0 - N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.8e+93], N[(1.0 - x), $MachinePrecision], N[(1.0 - N[(x - N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.15e91

   1. Initial program 99.7%

      \[\left(1 - x\right) + y \cdot \sqrt{x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto y \cdot \sqrt{x} + \color{blue}{\left(1 - x\right)} \]

      2. flip-+ N/A

         \[\leadsto \frac{\left(y \cdot \sqrt{x}\right) \cdot \left(y \cdot \sqrt{x}\right) - \left(1 - x\right) \cdot \left(1 - x\right)}{\color{blue}{y \cdot \sqrt{x} - \left(1 - x\right)}} \]

      3. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{y \cdot \sqrt{x} - \left(1 - x\right)}{\left(y \cdot \sqrt{x}\right) \cdot \left(y \cdot \sqrt{x}\right) - \left(1 - x\right) \cdot \left(1 - x\right)}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{y \cdot \sqrt{x} - \left(1 - x\right)}{\left(y \cdot \sqrt{x}\right) \cdot \left(y \cdot \sqrt{x}\right) - \left(1 - x\right) \cdot \left(1 - x\right)}\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(y \cdot \sqrt{x} - \left(1 - x\right)\right), \color{blue}{\left(\left(y \cdot \sqrt{x}\right) \cdot \left(y \cdot \sqrt{x}\right) - \left(1 - x\right) \cdot \left(1 - x\right)\right)}\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(y \cdot \sqrt{x}\right), \left(1 - x\right)\right), \left(\color{blue}{\left(y \cdot \sqrt{x}\right) \cdot \left(y \cdot \sqrt{x}\right)} - \left(1 - x\right) \cdot \left(1 - x\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \left(\sqrt{x}\right)\right), \left(1 - x\right)\right), \left(\color{blue}{\left(y \cdot \sqrt{x}\right)} \cdot \left(y \cdot \sqrt{x}\right) - \left(1 - x\right) \cdot \left(1 - x\right)\right)\right)\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{sqrt.f64}\left(x\right)\right), \left(1 - x\right)\right), \left(\left(y \cdot \color{blue}{\sqrt{x}}\right) \cdot \left(y \cdot \sqrt{x}\right) - \left(1 - x\right) \cdot \left(1 - x\right)\right)\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{\_.f64}\left(1, x\right)\right), \left(\left(y \cdot \sqrt{x}\right) \cdot \color{blue}{\left(y \cdot \sqrt{x}\right)} - \left(1 - x\right) \cdot \left(1 - x\right)\right)\right)\right) \]

      10. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{\_.f64}\left(\left(\left(y \cdot \sqrt{x}\right) \cdot \left(y \cdot \sqrt{x}\right)\right), \color{blue}{\left(\left(1 - x\right) \cdot \left(1 - x\right)\right)}\right)\right)\right) \]

      11. swap-sqr N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{\_.f64}\left(\left(\left(y \cdot y\right) \cdot \left(\sqrt{x} \cdot \sqrt{x}\right)\right), \left(\color{blue}{\left(1 - x\right)} \cdot \left(1 - x\right)\right)\right)\right)\right) \]

      12. rem-square-sqrt N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{\_.f64}\left(\left(\left(y \cdot y\right) \cdot x\right), \left(\left(1 - \color{blue}{x}\right) \cdot \left(1 - x\right)\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{\_.f64}\left(\left(x \cdot \left(y \cdot y\right)\right), \left(\color{blue}{\left(1 - x\right)} \cdot \left(1 - x\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \left(y \cdot y\right)\right), \left(\color{blue}{\left(1 - x\right)} \cdot \left(1 - x\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, y\right)\right), \left(\left(1 - \color{blue}{x}\right) \cdot \left(1 - x\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(\left(1 - x\right), \color{blue}{\left(1 - x\right)}\right)\right)\right)\right) \]

      17. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), \left(\color{blue}{1} - x\right)\right)\right)\right)\right) \]

      18. --lowering--.f64 35.5%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{\_.f64}\left(1, \color{blue}{x}\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 35.5%

      \[\leadsto \color{blue}{\frac{1}{\frac{y \cdot \sqrt{x} - \left(1 - x\right)}{x \cdot \left(y \cdot y\right) - \left(1 - x\right) \cdot \left(1 - x\right)}}} \]

   5. Taylor expanded in y around 0

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\color{blue}{\left(x - 1\right)}, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{\_.f64}\left(1, x\right)\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(x + \left(\mathsf{neg}\left(1\right)\right)\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, y\right)\right)}, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{\_.f64}\left(1, x\right)\right)\right)\right)\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(x + -1\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \color{blue}{\mathsf{*.f64}\left(y, y\right)}\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{\_.f64}\left(1, x\right)\right)\right)\right)\right) \]

      3. +-lowering-+.f64 5.2%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, y\right)\right)}, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{\_.f64}\left(1, x\right)\right)\right)\right)\right) \]

   7. Simplified 5.2%

      \[\leadsto \frac{1}{\frac{\color{blue}{x + -1}}{x \cdot \left(y \cdot y\right) - \left(1 - x\right) \cdot \left(1 - x\right)}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(1 + -1 \cdot \left(2 + {y}^{2}\right)\right)} \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot \left(1 + -1 \cdot \left(2 + {y}^{2}\right)\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(1 + -1 \cdot \left(2 + {y}^{2}\right)\right)}\right)\right) \]

      3. distribute-lft-in N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(1 + \left(-1 \cdot 2 + \color{blue}{-1 \cdot {y}^{2}}\right)\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(1 + \left(-2 + \color{blue}{-1} \cdot {y}^{2}\right)\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(1 + \left(\left(\mathsf{neg}\left(2\right)\right) + \color{blue}{-1} \cdot {y}^{2}\right)\right)\right)\right) \]

      6. associate-+r+ N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(1 + \left(\mathsf{neg}\left(2\right)\right)\right) + \color{blue}{-1 \cdot {y}^{2}}\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(1 + -2\right) + -1 \cdot {y}^{2}\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(-1 + \color{blue}{-1} \cdot {y}^{2}\right)\right)\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(-1 + \left(\mathsf{neg}\left({y}^{2}\right)\right)\right)\right)\right) \]

      10. unsub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(-1 - \color{blue}{{y}^{2}}\right)\right)\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \color{blue}{\left({y}^{2}\right)}\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \left(y \cdot \color{blue}{y}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 14.5%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right)\right)\right) \]

   10. Simplified 14.5%

       \[\leadsto \color{blue}{1 + x \cdot \left(-1 - y \cdot y\right)} \]

   if -3.15e91 < y < 1.8e93

   1. Initial program 100.0%

      \[\left(1 - x\right) + y \cdot \sqrt{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 - x} \]
   4. Step-by-step derivation

      1. --lowering--.f64 91.8%

         \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{x}\right) \]

   5. Simplified 91.8%

      \[\leadsto \color{blue}{1 - x} \]

   if 1.8e93 < y

   1. Initial program 99.7%

      \[\left(1 - x\right) + y \cdot \sqrt{x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto y \cdot \sqrt{x} + \color{blue}{\left(1 - x\right)} \]

      2. flip-+ N/A

         \[\leadsto \frac{\left(y \cdot \sqrt{x}\right) \cdot \left(y \cdot \sqrt{x}\right) - \left(1 - x\right) \cdot \left(1 - x\right)}{\color{blue}{y \cdot \sqrt{x} - \left(1 - x\right)}} \]

      3. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{y \cdot \sqrt{x} - \left(1 - x\right)}{\left(y \cdot \sqrt{x}\right) \cdot \left(y \cdot \sqrt{x}\right) - \left(1 - x\right) \cdot \left(1 - x\right)}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{y \cdot \sqrt{x} - \left(1 - x\right)}{\left(y \cdot \sqrt{x}\right) \cdot \left(y \cdot \sqrt{x}\right) - \left(1 - x\right) \cdot \left(1 - x\right)}\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(y \cdot \sqrt{x} - \left(1 - x\right)\right), \color{blue}{\left(\left(y \cdot \sqrt{x}\right) \cdot \left(y \cdot \sqrt{x}\right) - \left(1 - x\right) \cdot \left(1 - x\right)\right)}\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(y \cdot \sqrt{x}\right), \left(1 - x\right)\right), \left(\color{blue}{\left(y \cdot \sqrt{x}\right) \cdot \left(y \cdot \sqrt{x}\right)} - \left(1 - x\right) \cdot \left(1 - x\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \left(\sqrt{x}\right)\right), \left(1 - x\right)\right), \left(\color{blue}{\left(y \cdot \sqrt{x}\right)} \cdot \left(y \cdot \sqrt{x}\right) - \left(1 - x\right) \cdot \left(1 - x\right)\right)\right)\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{sqrt.f64}\left(x\right)\right), \left(1 - x\right)\right), \left(\left(y \cdot \color{blue}{\sqrt{x}}\right) \cdot \left(y \cdot \sqrt{x}\right) - \left(1 - x\right) \cdot \left(1 - x\right)\right)\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{\_.f64}\left(1, x\right)\right), \left(\left(y \cdot \sqrt{x}\right) \cdot \color{blue}{\left(y \cdot \sqrt{x}\right)} - \left(1 - x\right) \cdot \left(1 - x\right)\right)\right)\right) \]

      10. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{\_.f64}\left(\left(\left(y \cdot \sqrt{x}\right) \cdot \left(y \cdot \sqrt{x}\right)\right), \color{blue}{\left(\left(1 - x\right) \cdot \left(1 - x\right)\right)}\right)\right)\right) \]

      11. swap-sqr N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{\_.f64}\left(\left(\left(y \cdot y\right) \cdot \left(\sqrt{x} \cdot \sqrt{x}\right)\right), \left(\color{blue}{\left(1 - x\right)} \cdot \left(1 - x\right)\right)\right)\right)\right) \]

      12. rem-square-sqrt N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{\_.f64}\left(\left(\left(y \cdot y\right) \cdot x\right), \left(\left(1 - \color{blue}{x}\right) \cdot \left(1 - x\right)\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{\_.f64}\left(\left(x \cdot \left(y \cdot y\right)\right), \left(\color{blue}{\left(1 - x\right)} \cdot \left(1 - x\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \left(y \cdot y\right)\right), \left(\color{blue}{\left(1 - x\right)} \cdot \left(1 - x\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, y\right)\right), \left(\left(1 - \color{blue}{x}\right) \cdot \left(1 - x\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(\left(1 - x\right), \color{blue}{\left(1 - x\right)}\right)\right)\right)\right) \]

      17. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), \left(\color{blue}{1} - x\right)\right)\right)\right)\right) \]

      18. --lowering--.f64 26.0%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{\_.f64}\left(1, \color{blue}{x}\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 26.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{y \cdot \sqrt{x} - \left(1 - x\right)}{x \cdot \left(y \cdot y\right) - \left(1 - x\right) \cdot \left(1 - x\right)}}} \]

   5. Taylor expanded in y around 0

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\color{blue}{\left(x - 1\right)}, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{\_.f64}\left(1, x\right)\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(x + \left(\mathsf{neg}\left(1\right)\right)\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, y\right)\right)}, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{\_.f64}\left(1, x\right)\right)\right)\right)\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(x + -1\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \color{blue}{\mathsf{*.f64}\left(y, y\right)}\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{\_.f64}\left(1, x\right)\right)\right)\right)\right) \]

      3. +-lowering-+.f64 8.6%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, y\right)\right)}, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{\_.f64}\left(1, x\right)\right)\right)\right)\right) \]

   7. Simplified 8.6%

      \[\leadsto \frac{1}{\frac{\color{blue}{x + -1}}{x \cdot \left(y \cdot y\right) - \left(1 - x\right) \cdot \left(1 - x\right)}} \]

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\left(\frac{1}{x} + \frac{{y}^{2}}{x}\right) - 1\right)} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\frac{1}{x} + \frac{{y}^{2}}{x}\right) - 1\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\frac{1}{x} + \frac{{y}^{2}}{x}\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\frac{1}{x} + \frac{{y}^{2}}{x}\right) + -1\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(-1 + \color{blue}{\left(\frac{1}{x} + \frac{{y}^{2}}{x}\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \color{blue}{\left(\frac{1}{x} + \frac{{y}^{2}}{x}\right)}\right)\right) \]

      6. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \left(\frac{1}{x} + 1 \cdot \color{blue}{\frac{{y}^{2}}{x}}\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \left(\frac{1}{x} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{\color{blue}{{y}^{2}}}{x}\right)\right)\right) \]

      8. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \left(\frac{1}{x} - \color{blue}{-1 \cdot \frac{{y}^{2}}{x}}\right)\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \left(\frac{1}{x} - \frac{-1 \cdot {y}^{2}}{\color{blue}{x}}\right)\right)\right) \]

      10. div-sub N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \left(\frac{1 - -1 \cdot {y}^{2}}{\color{blue}{x}}\right)\right)\right) \]

      11. cancel-sign-sub-inv N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \left(\frac{1 + \left(\mathsf{neg}\left(-1\right)\right) \cdot {y}^{2}}{x}\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \left(\frac{1 + 1 \cdot {y}^{2}}{x}\right)\right)\right) \]

      13. *-lft-identity N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \left(\frac{1 + {y}^{2}}{x}\right)\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\left(1 + {y}^{2}\right), \color{blue}{x}\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left({y}^{2}\right)\right), x\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(y \cdot y\right)\right), x\right)\right)\right) \]

      17. *-lowering-*.f64 24.1%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, y\right)\right), x\right)\right)\right) \]

   10. Simplified 24.1%

       \[\leadsto \color{blue}{x \cdot \left(-1 + \frac{1 + y \cdot y}{x}\right)} \]

   11. Taylor expanded in x around 0

       \[\leadsto \color{blue}{1 + \left(-1 \cdot x + {y}^{2}\right)} \]
   12. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto 1 + \left(\left(\mathsf{neg}\left(x\right)\right) + {\color{blue}{y}}^{2}\right) \]

       2. associate-+r+ N/A

          \[\leadsto \left(1 + \left(\mathsf{neg}\left(x\right)\right)\right) + \color{blue}{{y}^{2}} \]

       3. sub-neg N/A

          \[\leadsto \left(1 - x\right) + {\color{blue}{y}}^{2} \]

       4. associate-+l- N/A

          \[\leadsto 1 - \color{blue}{\left(x - {y}^{2}\right)} \]

       5. --lowering--.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(x - {y}^{2}\right)}\right) \]

       6. --lowering--.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{\_.f64}\left(x, \color{blue}{\left({y}^{2}\right)}\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{\_.f64}\left(x, \left(y \cdot \color{blue}{y}\right)\right)\right) \]

       8. *-lowering-*.f64 24.2%

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right)\right) \]

   13. Simplified 24.2%

       \[\leadsto \color{blue}{1 - \left(x - y \cdot y\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 65.0% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.8 \cdot 10^{+93}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;1 - \left(x - y \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.8e+93) (- 1.0 x) (- 1.0 (- x (* y y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.8e+93) {
		tmp = 1.0 - x;
	} else {
		tmp = 1.0 - (x - (y * y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.8d+93) then
        tmp = 1.0d0 - x
    else
        tmp = 1.0d0 - (x - (y * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.8e+93) {
		tmp = 1.0 - x;
	} else {
		tmp = 1.0 - (x - (y * y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.8e+93:
		tmp = 1.0 - x
	else:
		tmp = 1.0 - (x - (y * y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.8e+93)
		tmp = Float64(1.0 - x);
	else
		tmp = Float64(1.0 - Float64(x - Float64(y * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.8e+93)
		tmp = 1.0 - x;
	else
		tmp = 1.0 - (x - (y * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.8e+93], N[(1.0 - x), $MachinePrecision], N[(1.0 - N[(x - N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.8e93

   1. Initial program 99.9%

      \[\left(1 - x\right) + y \cdot \sqrt{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 - x} \]
   4. Step-by-step derivation

      1. --lowering--.f64 74.0%

         \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{x}\right) \]

   5. Simplified 74.0%

      \[\leadsto \color{blue}{1 - x} \]

   if 1.8e93 < y

   1. Initial program 99.7%

      \[\left(1 - x\right) + y \cdot \sqrt{x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto y \cdot \sqrt{x} + \color{blue}{\left(1 - x\right)} \]

      2. flip-+ N/A

         \[\leadsto \frac{\left(y \cdot \sqrt{x}\right) \cdot \left(y \cdot \sqrt{x}\right) - \left(1 - x\right) \cdot \left(1 - x\right)}{\color{blue}{y \cdot \sqrt{x} - \left(1 - x\right)}} \]

      3. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{y \cdot \sqrt{x} - \left(1 - x\right)}{\left(y \cdot \sqrt{x}\right) \cdot \left(y \cdot \sqrt{x}\right) - \left(1 - x\right) \cdot \left(1 - x\right)}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{y \cdot \sqrt{x} - \left(1 - x\right)}{\left(y \cdot \sqrt{x}\right) \cdot \left(y \cdot \sqrt{x}\right) - \left(1 - x\right) \cdot \left(1 - x\right)}\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(y \cdot \sqrt{x} - \left(1 - x\right)\right), \color{blue}{\left(\left(y \cdot \sqrt{x}\right) \cdot \left(y \cdot \sqrt{x}\right) - \left(1 - x\right) \cdot \left(1 - x\right)\right)}\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(y \cdot \sqrt{x}\right), \left(1 - x\right)\right), \left(\color{blue}{\left(y \cdot \sqrt{x}\right) \cdot \left(y \cdot \sqrt{x}\right)} - \left(1 - x\right) \cdot \left(1 - x\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \left(\sqrt{x}\right)\right), \left(1 - x\right)\right), \left(\color{blue}{\left(y \cdot \sqrt{x}\right)} \cdot \left(y \cdot \sqrt{x}\right) - \left(1 - x\right) \cdot \left(1 - x\right)\right)\right)\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{sqrt.f64}\left(x\right)\right), \left(1 - x\right)\right), \left(\left(y \cdot \color{blue}{\sqrt{x}}\right) \cdot \left(y \cdot \sqrt{x}\right) - \left(1 - x\right) \cdot \left(1 - x\right)\right)\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{\_.f64}\left(1, x\right)\right), \left(\left(y \cdot \sqrt{x}\right) \cdot \color{blue}{\left(y \cdot \sqrt{x}\right)} - \left(1 - x\right) \cdot \left(1 - x\right)\right)\right)\right) \]

      10. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{\_.f64}\left(\left(\left(y \cdot \sqrt{x}\right) \cdot \left(y \cdot \sqrt{x}\right)\right), \color{blue}{\left(\left(1 - x\right) \cdot \left(1 - x\right)\right)}\right)\right)\right) \]

      11. swap-sqr N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{\_.f64}\left(\left(\left(y \cdot y\right) \cdot \left(\sqrt{x} \cdot \sqrt{x}\right)\right), \left(\color{blue}{\left(1 - x\right)} \cdot \left(1 - x\right)\right)\right)\right)\right) \]

      12. rem-square-sqrt N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{\_.f64}\left(\left(\left(y \cdot y\right) \cdot x\right), \left(\left(1 - \color{blue}{x}\right) \cdot \left(1 - x\right)\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{\_.f64}\left(\left(x \cdot \left(y \cdot y\right)\right), \left(\color{blue}{\left(1 - x\right)} \cdot \left(1 - x\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \left(y \cdot y\right)\right), \left(\color{blue}{\left(1 - x\right)} \cdot \left(1 - x\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, y\right)\right), \left(\left(1 - \color{blue}{x}\right) \cdot \left(1 - x\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(\left(1 - x\right), \color{blue}{\left(1 - x\right)}\right)\right)\right)\right) \]

      17. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), \left(\color{blue}{1} - x\right)\right)\right)\right)\right) \]

      18. --lowering--.f64 26.0%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{\_.f64}\left(1, \color{blue}{x}\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 26.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{y \cdot \sqrt{x} - \left(1 - x\right)}{x \cdot \left(y \cdot y\right) - \left(1 - x\right) \cdot \left(1 - x\right)}}} \]

   5. Taylor expanded in y around 0

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\color{blue}{\left(x - 1\right)}, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{\_.f64}\left(1, x\right)\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(x + \left(\mathsf{neg}\left(1\right)\right)\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, y\right)\right)}, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{\_.f64}\left(1, x\right)\right)\right)\right)\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(x + -1\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \color{blue}{\mathsf{*.f64}\left(y, y\right)}\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{\_.f64}\left(1, x\right)\right)\right)\right)\right) \]

      3. +-lowering-+.f64 8.6%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, y\right)\right)}, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{\_.f64}\left(1, x\right)\right)\right)\right)\right) \]

   7. Simplified 8.6%

      \[\leadsto \frac{1}{\frac{\color{blue}{x + -1}}{x \cdot \left(y \cdot y\right) - \left(1 - x\right) \cdot \left(1 - x\right)}} \]

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\left(\frac{1}{x} + \frac{{y}^{2}}{x}\right) - 1\right)} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\frac{1}{x} + \frac{{y}^{2}}{x}\right) - 1\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\frac{1}{x} + \frac{{y}^{2}}{x}\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\frac{1}{x} + \frac{{y}^{2}}{x}\right) + -1\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(-1 + \color{blue}{\left(\frac{1}{x} + \frac{{y}^{2}}{x}\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \color{blue}{\left(\frac{1}{x} + \frac{{y}^{2}}{x}\right)}\right)\right) \]

      6. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \left(\frac{1}{x} + 1 \cdot \color{blue}{\frac{{y}^{2}}{x}}\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \left(\frac{1}{x} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{\color{blue}{{y}^{2}}}{x}\right)\right)\right) \]

      8. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \left(\frac{1}{x} - \color{blue}{-1 \cdot \frac{{y}^{2}}{x}}\right)\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \left(\frac{1}{x} - \frac{-1 \cdot {y}^{2}}{\color{blue}{x}}\right)\right)\right) \]

      10. div-sub N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \left(\frac{1 - -1 \cdot {y}^{2}}{\color{blue}{x}}\right)\right)\right) \]

      11. cancel-sign-sub-inv N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \left(\frac{1 + \left(\mathsf{neg}\left(-1\right)\right) \cdot {y}^{2}}{x}\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \left(\frac{1 + 1 \cdot {y}^{2}}{x}\right)\right)\right) \]

      13. *-lft-identity N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \left(\frac{1 + {y}^{2}}{x}\right)\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\left(1 + {y}^{2}\right), \color{blue}{x}\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left({y}^{2}\right)\right), x\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(y \cdot y\right)\right), x\right)\right)\right) \]

      17. *-lowering-*.f64 24.1%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, y\right)\right), x\right)\right)\right) \]

   10. Simplified 24.1%

       \[\leadsto \color{blue}{x \cdot \left(-1 + \frac{1 + y \cdot y}{x}\right)} \]

   11. Taylor expanded in x around 0

       \[\leadsto \color{blue}{1 + \left(-1 \cdot x + {y}^{2}\right)} \]
   12. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto 1 + \left(\left(\mathsf{neg}\left(x\right)\right) + {\color{blue}{y}}^{2}\right) \]

       2. associate-+r+ N/A

          \[\leadsto \left(1 + \left(\mathsf{neg}\left(x\right)\right)\right) + \color{blue}{{y}^{2}} \]

       3. sub-neg N/A

          \[\leadsto \left(1 - x\right) + {\color{blue}{y}}^{2} \]

       4. associate-+l- N/A

          \[\leadsto 1 - \color{blue}{\left(x - {y}^{2}\right)} \]

       5. --lowering--.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(x - {y}^{2}\right)}\right) \]

       6. --lowering--.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{\_.f64}\left(x, \color{blue}{\left({y}^{2}\right)}\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{\_.f64}\left(x, \left(y \cdot \color{blue}{y}\right)\right)\right) \]

       8. *-lowering-*.f64 24.2%

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right)\right) \]

   13. Simplified 24.2%

       \[\leadsto \color{blue}{1 - \left(x - y \cdot y\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 65.3% accurate, 13.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 8 \cdot 10^{+131}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 8e+131) (- 1.0 x) (* y y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 8e+131) {
		tmp = 1.0 - x;
	} else {
		tmp = y * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 8d+131) then
        tmp = 1.0d0 - x
    else
        tmp = y * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 8e+131) {
		tmp = 1.0 - x;
	} else {
		tmp = y * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 8e+131:
		tmp = 1.0 - x
	else:
		tmp = y * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 8e+131)
		tmp = Float64(1.0 - x);
	else
		tmp = Float64(y * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 8e+131)
		tmp = 1.0 - x;
	else
		tmp = y * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 8e+131], N[(1.0 - x), $MachinePrecision], N[(y * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 7.9999999999999993e131

   1. Initial program 99.9%

      \[\left(1 - x\right) + y \cdot \sqrt{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 - x} \]
   4. Step-by-step derivation

      1. --lowering--.f64 70.9%

         \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{x}\right) \]

   5. Simplified 70.9%

      \[\leadsto \color{blue}{1 - x} \]

   if 7.9999999999999993e131 < y

   1. Initial program 99.8%

      \[\left(1 - x\right) + y \cdot \sqrt{x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto y \cdot \sqrt{x} + \color{blue}{\left(1 - x\right)} \]

      2. flip-+ N/A

         \[\leadsto \frac{\left(y \cdot \sqrt{x}\right) \cdot \left(y \cdot \sqrt{x}\right) - \left(1 - x\right) \cdot \left(1 - x\right)}{\color{blue}{y \cdot \sqrt{x} - \left(1 - x\right)}} \]

      3. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{y \cdot \sqrt{x} - \left(1 - x\right)}{\left(y \cdot \sqrt{x}\right) \cdot \left(y \cdot \sqrt{x}\right) - \left(1 - x\right) \cdot \left(1 - x\right)}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{y \cdot \sqrt{x} - \left(1 - x\right)}{\left(y \cdot \sqrt{x}\right) \cdot \left(y \cdot \sqrt{x}\right) - \left(1 - x\right) \cdot \left(1 - x\right)}\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(y \cdot \sqrt{x} - \left(1 - x\right)\right), \color{blue}{\left(\left(y \cdot \sqrt{x}\right) \cdot \left(y \cdot \sqrt{x}\right) - \left(1 - x\right) \cdot \left(1 - x\right)\right)}\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(y \cdot \sqrt{x}\right), \left(1 - x\right)\right), \left(\color{blue}{\left(y \cdot \sqrt{x}\right) \cdot \left(y \cdot \sqrt{x}\right)} - \left(1 - x\right) \cdot \left(1 - x\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \left(\sqrt{x}\right)\right), \left(1 - x\right)\right), \left(\color{blue}{\left(y \cdot \sqrt{x}\right)} \cdot \left(y \cdot \sqrt{x}\right) - \left(1 - x\right) \cdot \left(1 - x\right)\right)\right)\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{sqrt.f64}\left(x\right)\right), \left(1 - x\right)\right), \left(\left(y \cdot \color{blue}{\sqrt{x}}\right) \cdot \left(y \cdot \sqrt{x}\right) - \left(1 - x\right) \cdot \left(1 - x\right)\right)\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{\_.f64}\left(1, x\right)\right), \left(\left(y \cdot \sqrt{x}\right) \cdot \color{blue}{\left(y \cdot \sqrt{x}\right)} - \left(1 - x\right) \cdot \left(1 - x\right)\right)\right)\right) \]

      10. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{\_.f64}\left(\left(\left(y \cdot \sqrt{x}\right) \cdot \left(y \cdot \sqrt{x}\right)\right), \color{blue}{\left(\left(1 - x\right) \cdot \left(1 - x\right)\right)}\right)\right)\right) \]

      11. swap-sqr N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{\_.f64}\left(\left(\left(y \cdot y\right) \cdot \left(\sqrt{x} \cdot \sqrt{x}\right)\right), \left(\color{blue}{\left(1 - x\right)} \cdot \left(1 - x\right)\right)\right)\right)\right) \]

      12. rem-square-sqrt N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{\_.f64}\left(\left(\left(y \cdot y\right) \cdot x\right), \left(\left(1 - \color{blue}{x}\right) \cdot \left(1 - x\right)\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{\_.f64}\left(\left(x \cdot \left(y \cdot y\right)\right), \left(\color{blue}{\left(1 - x\right)} \cdot \left(1 - x\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \left(y \cdot y\right)\right), \left(\color{blue}{\left(1 - x\right)} \cdot \left(1 - x\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, y\right)\right), \left(\left(1 - \color{blue}{x}\right) \cdot \left(1 - x\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(\left(1 - x\right), \color{blue}{\left(1 - x\right)}\right)\right)\right)\right) \]

      17. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), \left(\color{blue}{1} - x\right)\right)\right)\right)\right) \]

      18. --lowering--.f64 12.0%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{\_.f64}\left(1, \color{blue}{x}\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 12.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{y \cdot \sqrt{x} - \left(1 - x\right)}{x \cdot \left(y \cdot y\right) - \left(1 - x\right) \cdot \left(1 - x\right)}}} \]

   5. Taylor expanded in y around 0

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\color{blue}{\left(x - 1\right)}, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{\_.f64}\left(1, x\right)\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(x + \left(\mathsf{neg}\left(1\right)\right)\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, y\right)\right)}, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{\_.f64}\left(1, x\right)\right)\right)\right)\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(x + -1\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \color{blue}{\mathsf{*.f64}\left(y, y\right)}\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{\_.f64}\left(1, x\right)\right)\right)\right)\right) \]

      3. +-lowering-+.f64 10.3%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, y\right)\right)}, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{\_.f64}\left(1, x\right)\right)\right)\right)\right) \]

   7. Simplified 10.3%

      \[\leadsto \frac{1}{\frac{\color{blue}{x + -1}}{x \cdot \left(y \cdot y\right) - \left(1 - x\right) \cdot \left(1 - x\right)}} \]

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\left(\frac{1}{x} + \frac{{y}^{2}}{x}\right) - 1\right)} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\frac{1}{x} + \frac{{y}^{2}}{x}\right) - 1\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\frac{1}{x} + \frac{{y}^{2}}{x}\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\frac{1}{x} + \frac{{y}^{2}}{x}\right) + -1\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(-1 + \color{blue}{\left(\frac{1}{x} + \frac{{y}^{2}}{x}\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \color{blue}{\left(\frac{1}{x} + \frac{{y}^{2}}{x}\right)}\right)\right) \]

      6. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \left(\frac{1}{x} + 1 \cdot \color{blue}{\frac{{y}^{2}}{x}}\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \left(\frac{1}{x} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{\color{blue}{{y}^{2}}}{x}\right)\right)\right) \]

      8. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \left(\frac{1}{x} - \color{blue}{-1 \cdot \frac{{y}^{2}}{x}}\right)\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \left(\frac{1}{x} - \frac{-1 \cdot {y}^{2}}{\color{blue}{x}}\right)\right)\right) \]

      10. div-sub N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \left(\frac{1 - -1 \cdot {y}^{2}}{\color{blue}{x}}\right)\right)\right) \]

      11. cancel-sign-sub-inv N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \left(\frac{1 + \left(\mathsf{neg}\left(-1\right)\right) \cdot {y}^{2}}{x}\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \left(\frac{1 + 1 \cdot {y}^{2}}{x}\right)\right)\right) \]

      13. *-lft-identity N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \left(\frac{1 + {y}^{2}}{x}\right)\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\left(1 + {y}^{2}\right), \color{blue}{x}\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left({y}^{2}\right)\right), x\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(y \cdot y\right)\right), x\right)\right)\right) \]

      17. *-lowering-*.f64 27.0%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, y\right)\right), x\right)\right)\right) \]

   10. Simplified 27.0%

       \[\leadsto \color{blue}{x \cdot \left(-1 + \frac{1 + y \cdot y}{x}\right)} \]

   11. Taylor expanded in y around inf

       \[\leadsto \color{blue}{{y}^{2}} \]
   12. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto y \cdot \color{blue}{y} \]

       2. *-lowering-*.f64 27.0%

          \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{y}\right) \]

   13. Simplified 27.0%

       \[\leadsto \color{blue}{y \cdot y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 61.3% accurate, 13.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0042:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x 0.0042) 1.0 (- 0.0 x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.0042) {
		tmp = 1.0;
	} else {
		tmp = 0.0 - x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 0.0042d0) then
        tmp = 1.0d0
    else
        tmp = 0.0d0 - x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 0.0042) {
		tmp = 1.0;
	} else {
		tmp = 0.0 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.0042:
		tmp = 1.0
	else:
		tmp = 0.0 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.0042)
		tmp = 1.0;
	else
		tmp = Float64(0.0 - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 0.0042)
		tmp = 1.0;
	else
		tmp = 0.0 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.0042], 1.0, N[(0.0 - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.00419999999999999974

   1. Initial program 99.9%

      \[\left(1 - x\right) + y \cdot \sqrt{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 - x} \]
   4. Step-by-step derivation

      1. --lowering--.f64 59.3%

         \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{x}\right) \]

   5. Simplified 59.3%

      \[\leadsto \color{blue}{1 - x} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   7. Step-by-step derivation

   8. Simplified 58.4%

      \[\leadsto \color{blue}{1} \]

   if 0.00419999999999999974 < x

   1. Initial program 99.9%

      \[\left(1 - x\right) + y \cdot \sqrt{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 - x} \]
   4. Step-by-step derivation

      1. --lowering--.f64 63.2%

         \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{x}\right) \]

   5. Simplified 63.2%

      \[\leadsto \color{blue}{1 - x} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{-1 \cdot x} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(x\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{x} \]

      3. --lowering--.f64 62.7%

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{x}\right) \]

   8. Simplified 62.7%

      \[\leadsto \color{blue}{0 - x} \]
   9. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(x\right) \]

      2. neg-lowering-neg.f64 62.7%

         \[\leadsto \mathsf{neg.f64}\left(x\right) \]

   10. Applied egg-rr 62.7%

       \[\leadsto \color{blue}{-x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0042:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0 - x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 31.7% accurate, 107.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 1.0)

C

double code(double x, double y) {
	return 1.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0
end function

Java

public static double code(double x, double y) {
	return 1.0;
}

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0;
end

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 99.9%

   \[\left(1 - x\right) + y \cdot \sqrt{x} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{1 - x} \]
4. Step-by-step derivation

   1. --lowering--.f64 61.4%

      \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{x}\right) \]

5. Simplified 61.4%

   \[\leadsto \color{blue}{1 - x} \]

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1} \]
7. Step-by-step derivation

8. Simplified 28.4%

   \[\leadsto \color{blue}{1} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024158 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, E"
  :precision binary64
  (+ (- 1.0 x) (* y (sqrt x))))